# Points A and B are at (5 ,8 ) and (7 ,3 ), respectively. Point A is rotated counterclockwise about the origin by pi  and dilated about point C by a factor of 5 . If point A is now at point B, what are the coordinates of point C?

Aug 4, 2018

$C = \left(- 8 , - \frac{43}{4}\right)$

#### Explanation:

$\text{under a counterclockwise rotation about the origin of } \pi$

• " a point "(x,y)to(-x,-y)

$A \left(5 , 8\right) \to A ' \left(- 5 , - 8\right) \text{ where A' is the image of A}$

$\vec{C B} = \textcolor{red}{5} \vec{C A '}$

$\underline{b} - \underline{c} = 5 \left(\underline{a} ' - \underline{c}\right)$

$\underline{b} - \underline{c} = 5 \underline{a} ' - 5 \underline{c}$

$4 \underline{c} = 5 \underline{a} ' - \underline{b}$

$\textcolor{w h i t e}{4 \underline{c}} = 5 \left(\begin{matrix}- 5 \\ - 8\end{matrix}\right) - \left(\begin{matrix}7 \\ 3\end{matrix}\right)$

$\textcolor{w h i t e}{4 \underline{c}} = \left(\begin{matrix}- 25 \\ - 40\end{matrix}\right) - \left(\begin{matrix}7 \\ 3\end{matrix}\right) = \left(\begin{matrix}- 32 \\ - 43\end{matrix}\right)$

$\underline{c} = \frac{1}{4} \left(\begin{matrix}- 32 \\ - 43\end{matrix}\right) = \left(\begin{matrix}- 8 \\ - \frac{43}{4}\end{matrix}\right)$

$\Rightarrow C = \left(- 8 , - \frac{43}{4}\right)$